In this write up we will be looking at tangent
circles. First let us look at the case that we have two circles
that are outside of each other. In this case the two given circles
are pink and the blue circle is tangent to both.

Next we will look at the case
that one of the two original circles is inside the other. Once
again, the pink circles are the originals and the blue circle
is the tangent circle.

Finally we will look at the case that the two
circles are overlapping. In this case the desired point of tangency
makes a big difference in the location of the tangent circle.
There are two pictures below. In one, the desired point of tangency
on circle A is located outside circle C so the tangent circle
is also outside circle C.

In the second case, the desired point of tangency
on circle A is inside circle C therefore the tangent circle is
inside of circle C and outside of circle A.

For a script to create a tangent circle to
two given circles click here.